Given information:
- The center of the circle is (4a-2, 6a^2).
- The circle passes through the point (-6, -2).
- The diameter of the circle is 40.

Approach:
To find the value of 'a', we can use the distance formula between two points. The distance between the center of the circle and the point (-6, -2) should be equal to half the diameter (20 units).

Solution:
Step 1: Write down the coordinates of the center of the circle and the given point.
Center of the circle: (4a-2, 6a^2)
Given point: (-6, -2)

Step 2: Apply the distance formula between the two points.
The distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Using the distance formula, we can write:
sqrt((-6 - (4a-2))^2 + (-2 - 6a^2)^2) = 20

Step 3: Simplify the equation and solve for 'a'.
Expand the squares and simplify the equation:
sqrt((10 - 4a)^2 + (-2 - 6a^2)^2) = 20
(10 - 4a)^2 + (-2 - 6a^2)^2 = 400

Expand the squares and simplify further:
(100 - 80a + 16a^2) + (4 + 24a^2 + 36a^4) = 400
36a^4 + 24a^2 - 80a + 120 + 16a^2 - 80a + 100 - 400 = 0
36a^4 + 40a^2 - 160a - 180 = 0

Step 4: Solve the quadratic equation.
We can solve the equation using factoring, completing the square, or the quadratic formula. Since the equation is a quadratic equation in terms of 'a', we can use factoring or the quadratic formula.

Step 5: Substitute the value of 'a' back into the equation.
Once we have solved the equation and found the values of 'a', we can substitute the values back into the original equations to verify if they satisfy the conditions.

Step 6: Finalize the answer.
After solving the quadratic equation and substituting the values of 'a' back into the original equations, we can determine the final value of 'a' that satisfies all the given conditions.

In this way, we can find the value of 'a' using the given information and the distance formula.